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CMSC 435
Introductory Computer Graphics
Rasterization Penny Rheingans
UMBC
Scan conversion
• Problem – How to generate filled polygons (by determining which
pixel positions are inside the polygon)
– Conversion from continuous to discrete domain
• Concepts – Spatial coherence
– Span coherence
– Edge coherence
Scanning Rectangles
for ( y from y0 to yn )
for ( x from x0 to xn )
Write Pixel (x, y, val)
Scanning Rectangles (2)
for ( y from y0 to yn )
for ( x from x0 to xn )
Write Pixel (x, y, val)
Scanning Rectangles (3)
for ( y from y0 to yn )
for ( x from x0 to xn )
Write Pixel (x, y, val)
Scanning Arbitrary Polygons • vertices:
(4, 1) , (7, 13) , (11 , 2)
Scanning Arbitrary Polygons (2) • vertices:
(4, 1) , (7, 13) , (11 , 2)
• Intersect scanline w/pgon edges => span extrema
Scanning Arbitrary Polygons (3) • vertices:
(4, 1) , (7, 13) , (11 , 2)
• Intersect scanline w/pgon edges => span extrema
• Fill between pairs of span extrema
Scanning Arbitrary Polygons (4) • vertices:
(4, 1) , (7, 13) , (11 , 2)
For each nonempty scanline
Intersect scanline w/pgon edges => span extrema
Fill between pairs of span extrema
Example Cases (2)
4 intersections w/ scanline 6 at x = 1, 6, 6, 12 1/7
Example Cases (3)
• 3 intersections w/scanline 5 at x = 1, 1, 11 5/7
Example Cases (4)
3 intersections w/scanline 5 at x = 1, 1, 11 5/7
==>
Count continuing edges once (shorten lower edge) now x=1, 11 5/7
Example Cases (5)
4 intersections w/ scanline 1at x = 5, 5, 10, 10
Example Cases (6)
4 intersections w/ scanline 1 at x = 5, 5, 10, 10
=>
Don't count vertices of horizontal edges.
Now x = 5, 10
Scanline Data Structures
Sorted edge table:
all edges
sorted by min y
holds:
max y
init x
inverse slope
Active edge table:
edges intersecting
current scanline
holds:
max y
current x
inverse slope
Scanline Algorithm 1. Bucket sort edges into sorted edge table
2. Initialize y & active edge table
y = first non- empty scanline
AET = SET [y]
3. Repeat until AET and SET are empty
Fill pixels between pairs of x intercepts in AET
Remove exhausted edges
Y++
Update x intercepts
Resort table (AET)
Add entering edges
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
sort on minY: 1
store:
max Y: 11
min X: 4
1/m : (Xmax - Xmin) / (Ymax - Ymin) = (1 - 4) / (11 - 1) = -3 / 10
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
initialize active edge list to first non empty scanline
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
initialize active edge list to first non empty scanline
for each non empty scanline
fill between pairs (x=4,12)
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
initialize active edge list to first non empty scanline
for each non empty scanline
fill between pairs (x=4,12)
remove exhausted edges
update intersection points
resort table
add entering edges
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
initialize active edge list to first non empty scanline
for each non empty scanline
fill between pairs (x=3 1/10,12)
remove exhausted edges
update intersection points
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
initialize active edge list to first non empty scanline
for each non empty scanline
fill between pairs (x=3 1/10,12)
remove exhausted edges
update intersection points
resort table
add entering edges
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
initialize active edge list to first non empty scanline
for each non empty scanline
fill between pairs (x = 2 8/10, 9; 9,12)
remove exhausted edges
update intersection points
resort table
add entering edges
Example: vertices (4,1), (1,11), (9,5), (12,8), (12,1)
bucket sort edges into sorted edge table
initialize active edge list to first non empty scanline
for each non empty scanline
fill between pairs (x=2 5/10, 7 2/3; 10,12)
remove exhausted edges
update intersection points
resort table
add entering edges
Fill Variants
Fill between pairs:
for ( x = x1; x < x2; x++ )
framebuffer [ x, y ] = c
Fill Variants (2)
• Pattern Fill
Fill between pairs:
for ( x = x1; x < x2; x++ )
if ( ( x + y ) % 2 )
framebuffer [ x, y ] = c1
else
framebuffer [ x, y ] = c1
Fill Variants (3)
• Colorwash
Red to blue
Fill between pairs:
for ( x = x1; x < x2; x++ ) framebuffer [ x, y ] = C0 + dC * ( x1 - x )
For efficiency carry C and dC in AETand calculate color incrementally
Fill Variants (4)
• Vertex colors
Red, green, blue
Fill between pairs:
for ( x = x1; x < x2; x++ ) framebuffer [ x, y ] =
Cy1x1 + [(x - x1)/(x2 - x1)*(Cy1x2 - Cy1x1)]/dCx
For efficiency carry Cy and dCy in AET calculate dCx at beginning of scanline
Barycentric Coordinates
• Use non-orthogonal coordinates to describe
position relative to vertices
– Coordinates correspond to scaled signed
distance from lines through pairs of vertices
p = a+ b a( ) + c a( ) p , ,( ) = a+ b+ c
Barycentric Example
Barycentric Coordinates
• Computing coordinates
=ya yb( )x+ xb xa( )y+ xayb xbyaya yb( )xc + xb xa( )yc + xayb xbya
=ya yc( )x+ xc xa( )y+ xayc xbyaya yc( )xb + xc xa( )yb + xayc xcya
=1
Alternative Computation
Barycentric Rasterization For all x do
For all y do
Compute ( , , ) for (x,y)
If ( [0,1] and [0,1] and [0,1] then
c = c0 + c1 + c2Draw pixel (x,y) with color c
Barycentric Rasterization xmin = floor(xi)
xmax = ceiling(xi)
ymin = floor(yi)
ymax = ceiling(xi)
for y = ymin to ymax do
for x = xmin to xmax do
= f12(x,y)/f12(x0,y0)
= f20(x,y)/f20(x1,y1)
= f01(x,y)/f01(x2,y2)
If ( [0,1] and [0,1] and [0,1] then
c = c0 + c1 + c2Draw pixel (x,y) with color c
Barycentric Rasterization
• Computing coordinates
=f01(x, y)
f01(x2, y2 )=
y0 y1( )x+ x1 x0( )y+ x0y1 x1y0y0 y1( )x2 + x1 x0( )y2 + x0y1 x1y0
=f20 (x, y)
f20 (x1, y1)=
y2 y0( )x+ x0 x2( )y+ x2y0 x0y2y2 y0( )x1 + x0 x2( )y1 + x2y0 x0y2
=f12 (x, y)
f12 (x0, y0 )=
y1 y2( )x+ x2 x1( )y+ x1y2 x2y1y1 y2( )x0 + x2 x1( )y0 + x1y2 x2y1
